Matrices & Determinants Questions and Answers

Solve the matrix equation for X: -1 0 1 1 3 0 X 1 1 0 = -9 1 12 3 1 -1

If f(x) = 3x + 2 and g(x)= 1/3(x-1) what is the value of f[g(4)]? A 5 B.-3 C= 50/3 D.-4

All solutions [x, y, z] of the linear equation 9x- 25y + 24z = -20 geometrically represent a plane P in R³. Find a normal vector n to P and the position vectors p, q, r of three distinct points in P:

What is the initial amount for the function: f(x) = 300(1.16)^x (A) 300 (B) 16 (C) 1.16 (D) x

Let be: A := (0 -1) ∈ R²,² (1 -1) determine A²⁰²².

Given a matrix A = ( 2 5, -1 5, 0 1) determine matrices Q and R such that A= QR.

Let be the vector from initial point P₁ = (-2,15) to terminal point P₂ = (7,8). Write v in terms of i and j .

Evaluate A = {P(1 + r)^t} / {(1 + r)^2} Click on the given link to view the question. Evaluate A when P = $9000, r = 0.05, t = 6 years. Do enter any units in your answer. A=

Find the transition matrix from B' to B Find [x]B , when provided with [x]B B = {(1, 1, 1), (1, -1, 1), (0, 0, 1)}, B' = {(2, 2, 0), (0, 1, 1), (1, 0, 1)), [X]B = 2 3 1

If A is a 3 x 3 matrix with three orthogonal eigenvectors, then A is diagonalizable. True False

Solve the system of equations. If the system has an infinite number of solutions, express them in terms of the parameter z. 9x + 8y - 42z = 6 4x + 7y - 29z = 13 x + 2y - 8z = 4 x = y = z =

Factor by grouping: 7x³ - 8x² - 42x - 48

A local zoo has just opened a new stingray environment with 7 young, healthy stingrays. The population of stingrays in the enclosure is expected to at least double every year and can be represented after x years by this inequality. y ≥ 7(2)ˣ At the time the stingrays were added to the enclosure, the zookeepers determined that the enclosure could sustain at most 160 stingrays. The number of stingrays the enclosure can sustain after x years, because of budget constraints, can be represented by this inequality. y≤ 160-2x Which point on the graph represents a possible number of stingrays, y, in the enclosure?

Kevin, Kim, and Lyu are walking on a 2D surface embedded in a flat three-dimensional Euclidean space as z = r cos Φ y = rsinΦ, z=2/3r^3/2 Show that the induced metric on the two-dimensional surface is

All solutions [x, y, z] of the linear equation 9x - 25y + 24z = -20 geometrically represent a plane P in R³. Find a normal vector n to P and the position vectors p, q, r of three distinct points in P:

The Routh Criterion Stability S(S^2+8S+a)+4(S+8)=0 solution ...control systems

Let W be the set of all vectors [x y x+y] with x and y real. Find a basis of W¹.

The rate of change of the number of coyotes in a population is directly proportional to N(t), where t is the time in years. a.) Set up a differential equation and find the general solution containing C and k. b.) Find the particular solution to the above differential equation given the coyote population increases from 300 to 500 in two years.

Let R be a ring. On which of the following sets is matrix multiplication a well-defined operation? Select all correct answers. 1. The set of all matrices of all sizes with entries in R 2. The set of all 2x2 matrices with entries in R 3. The set of all 2x3 matrices with entries in R

Use the limit definition of the derivative and the four-step process to find f'(x). f(x)=x²+ 5x-6 f'(x)=

Solve the system of linear equations algebraically. Show/explain all steps in an organized manner. x+y+z=1 -2x+y+z= -2 3x + 6y + 6z = 5

Given that W is the subspace of R^4 spanned by the vectors (1,0,-1,2), (-1,2,1,1), (1,5,0,1) find a basis for the orthogonal complement of W

Find an orthogonal matrix P that diagonalizes the matrix A = 8 -2 2 -2 5 4 2 4 5

A polyhedron is a figure whose surfaces are b. A net is a pattern that can be folded to form a figure.

A local movie theater is trying to find the best price at which to sell popcorn. To reach its goal of making at least $50,000 from popcorn sales this year, the theater decided to hire a consulting firm to analyze its business. The firm determined that the best-case scenario for the theater's revenue generated from popcorn sales, while meeting its revenue goals, is given by this system of inequalities, where r represents the revenue in tens of thousands of dollars and p represents the sale price of popcorn in dollars. r≤-0.23² +2.25 p r⋝5 Complete the statements about the system's possible solutions. The point (4,6) is_____ of this system. The point (6,5) is_____ of this system.

Which two variables are most likely to have a positive correlation? a)The length of a person's hair and the amount of shampoo used b)A person's age and the quality of their eyesight c)The average number of hours spent watching TV daily and the age of the TV

Assume that T and S are matrix of the same size. Prove or Disprove that (T+ S)² is a symmetric, skew- symmetric or neither.

Solve the following system of linear equations using the LU factorization method. x₁ + x₂ + 3x4 = 7 2x₁ + x₂-x3+ x4 = 5 3x₁-x₂-x₂ + 2x4 = 8 -x₁ + 2x₂ + 3x₂ - x4 = -6

Martin made a net of a rectangular prism with a square cardboard of side 16 cm as shown in the figure. He then cut and folded the cardboard to make the prism. (a) What was the total surface area of the prism? (b) What was the volume of the prism?

If logb²=0.43 and log b³=0.68, evaluate the following. log b⁸ log b⁸=

Problem 4. For each of the following conditions, either draw a simple graph with the required conditions, or show that no such graph can exist. (a) 6 vertices, 4 edges. (b) 5 vertices with degrees 1, 2, 2, 3, 4. (c) 6 vertices with degrees 1, 1, 2, 3, 4, 4. (d) 6 vertices with degrees 1, 1, 3, 4, 4, 5.

X = 8. Goes through the origin. Has a normal vector parallel to the xy-plane. Goes through the point (0, 8, 0). Has a normal vector whose dot products withi,j,k are all positive.

Diagonalize these Hermitian matrices to reach S = QAQH:

Consider the symmetric positive definite matrix A = Try to find a set of directions which are conjugate about matrix A.

For the matrices A and B given in Question 3, find BA if possible. a) Not possible. b) -8 16 -7 9 -3 5 c) -4 8 -3 3 1 1 A=-2 2 -1 3 B= 2 4 3 1

Solve the system of two linear inequalities graphically. { x > -4 { y ≥ 1

Give a geometric description of Span (v₁ , V₂) for the vectors (A) Span (v₁, v₂) cannot be determined with the given information. (B) Span (v₁ , v₂) is the plane in R³ that contains V₁, V₂, and 0. (C) Span (v₁ , v₂) is the set of points on the line through v, and 0. (D) Span (v₁ , v₂) is R³.

Solve using Gaussian or Gauss-Jordan elimination. W+ x + y + z = 5 3w+ 3x-3y 3z=9 4w 3x + 3y + Z=14 W-x + 5y + 4z = 8 (A). The solution is (B). There are infinitely many solutions. The solutions are where z is any real number. (C). There is no solution.

Find a matrix P that diagonalizes A, and compute p-¹AP.

Consider the matrix A = 3 (a) Compute the adjoint matrix adj(A). (b) Compute the determinant det(A) and find the inverse matrix A-¹, Consider the matrix B= 2 1 (a) Use the cofactor exapansion to evaluate the determinant det(B) with using the second column. (b) Use the cofactor exapansion to evaluate the determinant det(B) with using the third row.

Compute the product of the following partitioned matrices using block multiplication.

Let T be a linear transformation from R² to R² defined by T(x,y) = (2x -y, x + y). a) Find the matrix of T with respect to the bases B₁ = {(1.1). (2,1)} and B₂ = {(-1,2), (1,1)). b) Use the matrix found in part a) to find T(v), where v = (2.3).

Compute the eigenvalues of A using un-shifted QR algorithm. Can you get improvement by using the shifted version of QR algorithm? Show your work.

Use the inner product (p, q) = aobo + a₁b₁ + a2b₂ to find (p, q), ||pl|, |la||, and d(p, q) for the polynomials in P2. p(x) = = 3- x + 2x², g(x) = x - x²

Define the linear transformation T by T(x) = Ax. Find ker(7), nullity(7), range(7), and rank(7).

Let A = {a₁a2 a3} and B = {b₁ b2b3} be bases for a vector space V, and suppose a₁ = 3b₁-b₂, a2 = -b₁ + b₂ + b3, a3 = b₂ - 4b3- a. Find the change-of-coordinates matrix from A to B. b. Find [x]g for x =2a₁ + a₂ + a3.

Prove that if the matrix A has orthogonal columns, then Az = b has a unique least squares solution, and find this unique solution.

Change each degree measure to radian measure in terms of π 1) 135° 2) 300° 3) 210°

Find the first & second derivatives of the following functions: a) f(x) = Q^(⅓) b) f(x)=(Y⁴-1)/Y⁴

Let A=(¹₋₁ ⁻¹ ₁ ²₋₂). Find the singular value decomposition (SVD) of A